Methods, systems, and apparatus for reducing cogging torque in an electric machine

ABSTRACT

An interior permanent magnet machine is described. The machine includes a rotor rotatable about a central machine axis. The rotor includes a plurality of permanent magnet openings and a plurality of permanent magnets disposed therein. The permanent magnet openings are separated by rotor webs configured to facilitate reducing leakage flux through the rotor webs. The machine also includes a stator disposed coaxially with the rotor and separated from the rotor by a circumferential air gap. The stator includes a plurality of stator teeth that define a plurality of stator slots therebetween. The stator teeth include a stator tooth tip configured to facilitate reducing cogging torque and torque ripple.

BACKGROUND OF THE INVENTION

The subject matter described herein relates generally to electricmachines, and more specifically, to methods, systems, and apparatus forreducing cogging torque in an electric machine.

Electric machines, including electric generators and electric motors,are used in countless varieties and applications worldwide. Typically,the rotational force and torque generated within an electric motor isdelivered to an application by a rotor shaft. The torque generated is aproduct of current applied to the motor and an electromagnetic fieldmaintained in the motor. This delivered torque varies as a function oftime and position of the rotor. Magnets on the rotor generate a rotormagnetic field and current on a stator winding generates a statormagnetic field. When the rotor generated magnetic field approaches thestator generated magnetic field, the torque is positive, and when therotor magnetic field leaves the stator magnetic field the torque isnegative. The torque produced is therefore non-uniform and known tothose in the art as torque ripple. A second component of non-uniformtorque is known as cogging torque. Cogging torque is present because therotor magnets prefer to line up with the stator teeth. In someapplications, the ripple and/or cogging torque produces objectionablevibration at the motor shaft resulting in end product noise.Furthermore, ripple and/or cogging torque may produce undesirable statortorsional and/or radial forces

One example of such an application occurs when a motor drives an endproduct, for example, a fan. Cogging torque produces vibrations whichare transmitted to machine components such as the motor and fanmounting. These vibrations produce undesirable noise as the coggingfrequencies couple with ‘application’ resonances. In addition toacoustic noise, continued exposure over time to such vibrations mayloosen motor and fan assemblies, and ultimately may cause a motorfailure. Isolation and damping systems, for example, an isolated rotor,may be employed to minimize the effects of the vibrational energyinduced into the motor and fan system.

Furthermore, in applications that include a rotor that includespermanent magnets, such as a brushless direct current (BLDC) motor or abrushless alternating current (BLAC) motor, a resultant noise due toinherent cogging torque is caused by rotor permanent magnets moving paststator teeth. Cogging torque may be reduced by including a skewedmagnetic field. However, it is currently difficult to apply a skewedmagnetic field in a motor that includes an interior permanent magnetrotor. Adding sub-slots to the stator teeth will reduce the compositepeak-to-peak cogging torque, but adds higher frequency components thatcan excite system resonances in some applications. Cogging torque atfrequencies other than the fundamental frequency may also cause motorvibration and generate noise in an end product. Furthermore, coggingtorque may be reduced by using a resilient rotor construction. However,a resilient rotor significantly increases a cost of the motor, whileadding a potential failure mechanism to the motor. In addition, theremay also be undesirable interactions between the magnetic fieldgenerated by the stator windings and the back EMF of the PM motor thatmay create torque pulsations (i.e. torque ripple) rich with harmoniccontent.

Moreover, efficiency of BLDC and BLAC motors with permanent magnetsembedded in the rotor (e.g., an interior permanent magnet rotor) istypically limited due to leakage flux of individual permanent magnetsthrough the rotor core.

BRIEF DESCRIPTION OF THE INVENTION

In one aspect, an interior permanent magnet machine is provided. Themachine includes a rotor rotatable about a central machine axis. Therotor includes a plurality of permanent magnet openings and a pluralityof permanent magnets disposed therein. The permanent magnet openings areseparated by rotor webs configured to facilitate reducing leakage fluxthrough the rotor webs. The machine also includes a stator disposedcoaxially with the rotor and separated from the rotor by acircumferential air gap. The stator includes a plurality of stator teeththat define a plurality of stator slots therebetween. The stator teethinclude a stator tooth tip configured to facilitate reducing coggingtorque.

In another aspect, a stator for use in an electric machine is provided.The stator includes an outer ring concentric to an axial center line ofthe electric machine. The stator also includes a plurality of statorteeth extending radially inward from the outer ring toward the axialcenter line. Each of the plurality of stator teeth includes a statortooth tip having a first surface. The stator tooth tip includes a firstend, a second end, and a center section therebetween. The first end andthe second end are tapered away from the axial center line such that adistance between the first surface and the axial center line is greaterat the first end and the second end than at the center section.

In yet another aspect, a method for reducing cogging torque in anelectric machine that includes an interior permanent magnet rotor thatis concentric to a stator is provided. The stator includes a pluralityof stator teeth each including a stator tooth tip. The method includesdetermining an angle of a first end of the stator tooth tip relative toa center section of the stator tooth tip that results in a graded airgap when assembled with the rotor, the graded air gap reduces coggingtorque and changes cogging frequency favorably to reduce noise in theelectric machine.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a portion of an exemplary electric motor.

FIG. 2 is a cross sectional view of an exemplary embodiment of a motorassembly that may be included within the motor shown in FIG. 1.

FIG. 3 is a cross sectional view of a portion of a stationary assemblythat may be included within the motor shown in FIG. 1.

FIG. 4 is a cross sectional view of a stator tooth shown in FIG. 2.

FIG. 5 is a cross sectional view of a portion of a rotatable assemblythat may be included within the motor shown in FIG. 1.

FIG. 6 is a flow chart of an exemplary method for reducing noise in anelectric motor.

DETAILED DESCRIPTION OF THE INVENTION

The methods, systems, and apparatus described herein facilitate reducingnoise in an electric motor and its intended application withoutincreasing manufacturing cost or complexity, and with minimal adverseeffects on motor efficiency when compared to other options. The methods,systems, and apparatus described herein also facilitate reducingmotor-to-motor variation effect on noise. For example, a stator toothincluded within a stator lamination used in an electric motor may beshaped in such a way that an amplitude of cogging torque at keyfrequencies (fundamental and harmonics of various cogging sources) isreduced so that noise in the application is reduced. In addition, thestator tooth shape may be shaped to reduce higher order back EMFharmonics effecting torque ripple generated primarily by the currentapplied to the stator windings. Furthermore, a magnet positioningopening in a lamination of an interior permanent magnet rotor may besized and shaped in such a way that the leakage flux between north andsouth poles of each permanent magnet is minimized, and yet laminationsare easily manufactured, and rotor-to-rotor variation is reduced.Reducing leakage flux facilitates increasing motor efficiency.

FIG. 1 is an illustration of a portion of an exemplary electric motor10. Motor 10 may include an electronically commutated motor (ECM), abrushless direct current (BLDC) motor, a brushless alternating current(BLAC) motor, or any other suitable electric motor. Furthermore,although illustrated in conjunction with BLAC motor 10, the methods,systems, and apparatus described herein are not limited to use with onlyBLAC motor 10, rather, the methods, systems, and apparatus describedherein may be applied to any suitable electric motor. Motor 10 includesa motor assembly housing 12. Motor 10 includes a motor assembly 14 thatincludes a stationary assembly 16 and a rotatable assembly 18. Motorassembly housing 12 defines an interior 22 and an exterior 24 of motor10 and is configured to at least partially enclose and protect motorassembly 14. Stationary assembly 16 includes a stator core 28, whichincludes a plurality of stator teeth 30 and a plurality of windings 32wound around stator teeth 30. In an exemplary embodiment, stationaryassembly 16 is a three phase salient pole stator assembly, stator core28 is formed from a stack of laminations made of a highly magneticallypermeable material, and windings 32 are wound on stator core 28 in amanner known to those of ordinary skill in the art. Rotatable assembly18 includes a permanent magnet rotor core 36 and a shaft 38. In theexemplary embodiment, rotor core 36 is formed from a stack oflaminations made of a magnetically permeable material and issubstantially received in a central bore of stator core 28. Rotor core36 and stator core 28 are illustrated as being solid in FIG. 1 forsimplicity, their construction being well known to those of ordinaryskill in the art. While FIG. 1 is an illustration of a three phasemotor, the methods and systems described herein may be applied to motorshaving a single phase or multiple phases.

In the exemplary embodiment, motor 10 also includes an endshield 42coupled to motor assembly housing 12. Endshield 42 includes an opening44 configured to support shaft 38 of rotatable assembly 18. Endshield 42is also configured to secure stationary assembly 16 and rotatableassembly 18 within motor assembly housing 12. In the exemplaryembodiment, motor 10 is included within an air handling system. Forexample, motor 10 may be coupled to a fan (not shown) for moving airthrough an air handling system, for blowing air over cooling coils,and/or be coupled to an air conditioning compressor for driving thecompressor. More specifically, motor 10 may be used in air movingapplications used in the HVAC industry, for example, in residentialapplications using ⅓ horsepower to 1 horsepower motors and/or incommercial and industrial applications and hermetic compressor motorsused in air conditioning applications. Although described herein in thecontext of an air handling system, motor 10 may engage any suitable workcomponent and be configured to drive such a work component.

FIG. 2 is a cross sectional view of an exemplary embodiment of a motorassembly, for example, motor assembly 14 (shown in FIG. 1) that may beincluded within motor 10 (shown in FIG. 1). As described above, motorassembly 14 includes a stationary assembly 16 and a rotatable assembly18. FIG. 3 is a cross sectional view of a portion 50 of stationaryassembly 16 (shown in FIG. 2). FIG. 4 is a cross sectional view of astator tooth that may be included in stationary assembly 16 (shown inFIG. 2). FIG. 5 is a cross sectional view of a portion 52 of rotatableassembly 18 (shown in FIG. 2).

Stationary assembly 16 includes stator core 28. In the exemplaryembodiment, stator core 28 includes a plurality of stator teeth, forexample stator teeth 60, 62, and 64, that extend from an outer ring 66.Outer ring 66 is concentric about an axial center line 68 of motor 10.As described above, windings 32 (shown in FIG. 1) are wound aroundstator teeth 60, 62, and 64. A plurality of stator slots, for example,stator slots 72 and 74 are defined between stator teeth 60, 62, and 64.Each of the plurality of stator teeth includes a tooth tip. For example,stator tooth 60 includes a stator tooth tip 80, stator tooth 62 includesa stator tooth tip 82, and stator tooth 64 includes a stator tooth tip84. Stator tooth tips 80, 82, and 84 each include an interior surface88.

In the exemplary embodiment, interior surface 88 of stator tooth tips80, 82, and 84 are shaped to reduce the amplitude of cogging torque atspecific frequencies so that acoustical noise in the application wheremotor 10 is used is reduced. In the exemplary embodiment, the amplitudeof cogging torque at the commonly defined or native cogging torquefrequency is reduced while also reducing the amplitude of the coggingtorque increase at other frequencies, for example, a pole pass frequencyand/or a slot order frequency. In the specific embodiment of motor 10,which includes a ten pole rotor and twelve stator slots, the fundamentalfrequency of the native cogging torque is equal to sixty times therevolutions per minute (RPM) frequency of motor 10 (e.g., the leastcommon multiple of ten and twelve). Furthermore, the cogging torque willinclude other frequencies, for example, at a pole pass frequency and aslot order frequency with amplitude dependent on motor design and normalmotor-to-motor manufacturing variances. In this example, the pole passfrequency is equal to ten times the RPM frequency and the slot orderfrequency is equal to twelve times the RPM frequency. Multiples of allof these cogging sources (harmonics) are also common with amplitudes ofall frequencies dependent on motor design. In the exemplary embodiment,interior surface 88 is shaped such that the amplitude of the coggingtorque at the native frequency is reduced with minimal increase in theamplitude of the fundamental cogging torque at other frequencies,including the pole pass frequency and/or the slot order frequency. Inaddition, the tooth shape reduces and/or eliminates the higher harmonicfrequencies of the stator winding back EMF interacting with the appliedstator electromagnetic field, which provides a reduction inelectromagnetic induced torque ripple.

Rotatable assembly 18 is substantially received in a central bore ofstationary assembly 16 such that an outer surface 90 of rotor core 36 isseparated from interior surface 88 of stator teeth 60, 62, and 64 by anair gap 96. Stator teeth 60, 62, and 64 are configured to facilitatereducing an amplitude of the cogging torque of motor 10. Morespecifically, interior surface 88 of stator tooth tips 80, 82, and 84have a cross-sectional profile that causes air gap 96 to change duringoperation of motor 10. Cogging torque naturally exists in an electricmotor because the magnetic energy that is stored in air gap 96 variesaccording to an angular position of rotor core 36 with respect tostationary assembly 16. Due to a shape of interior surface 88 of toothtips 80, 82, and 84, air gap 96 also varies according to a distancebetween interior surface 88 and outer surface 90. More specifically, abackward curvature of portions of stator tooth tip 80 increases air gap96 at ends of stator tooth tip 80 as compared to air gap 96 at a centerof stator tooth 60 resulting in a graded air gap. The graded air gapfacilitates reducing cogging torque in such a way that noise issuppressed in the end product. The air gap 96 is varied enough toeffectively reduce the amplitude of cogging torque at the fundamentalfrequency of motor 10, but not enough to introduce additional noiseissues to motor 10 and the end product. In the exemplary embodiment, airgap 96 is varied such that the amplitude of the native cogging torquefundamental is reduced, and the higher harmonic back EMF is reduced,with reduced effect on fundamental cogging torque increase at the polepass and slot order frequencies.

In the exemplary embodiment, stator tooth tip 80 includes a first end100, a second end 102, and a center section 104. First end 100 islocated a first distance 110 in a first circumferential direction 120from center section 104 of stator tooth tip 80. Second end 102 islocated a second distance 122 in a second circumferential direction 124from center section 104 of stator tooth tip 80, wherein firstcircumferential direction 120 is opposite to second circumferentialdirection 124. A first line 126 illustrates a division between first end100 and center section 104. Line 126 extends between a center 128 ofstationary assembly 16 through a point on interior surface 88 wherecenter section 104 transitions to first end 100. A second line 130similarly illustrates a division between second end 102 and centersection 104.

In the exemplary embodiment, air gap 96 is larger between interiorsurface 88 of first end 100 and outer surface 90 than between interiorsurface 88 of center section 104 and outer surface 90. Similarly, airgap 96 is larger between interior surface 88 of second end 102 and outersurface 90 than between interior surface 88 of center section 104 andouter surface 90. In the exemplary embodiment, interior surface 88 offirst end 100 is angled at an angle α with respect to center section 104such that air gap 96 between rotor core 36 and stator tooth tip 80increases as a circumferential distance from center section 104increases. In an alternative embodiment, interior surface 88 of firstend 100 includes a first curve configured such that air gap 96 increasesas a circumferential distance from center section 104 increases. Angle αof first end 100 relative to center section 104, and/or the curve offirst end 100, is determined based at least partially on a frequencycontent of the cogging torque wave generated by motor 10. Second end 102is configured substantially similarly to first end 100.

FIG. 4 is a cross sectional view of a stator tooth, for example, statortooth 60 (shown in FIG. 3). A first angle θ is an angle measured betweenfirst line 126 (shown in FIG. 2) and a third line 132. First line 126extends between center 128 of stationary assembly 16 to an intersectionbetween center section 104 and first end 100. Third line 132 extendsbetween center 128 and a point equidistant from first end 100 of statortooth 60 and an adjacent stator tooth. A radial distance 136 is definedas a maximum distance between interior surface 88 and a circle centeredat center 128 of a diameter equal to a distance from center 128 tointerior surface 88 of center section 104. Radial distance 136 may alsobe described as a difference between a maximum air gap 96 and a minimumair gap 96 over tooth tip 80. A distance 138 is a thickness of centersection 104. Angle θ and distance 136 are optimized to reduce coggingtorque and third harmonic content of the cogging torque. In theexemplary embodiment, angle θ corresponds to a number of stator teethand rotor poles. For example, angle θ may be determined by:

${0 < \theta \leq \frac{360^{\circ}}{2n}},$

where n=number of stator teeth. Furthermore, in the exemplaryembodiment, radial distance 136 is less than distance 138.

Referring now to FIG. 5, in the exemplary embodiment, rotor core 36includes a plurality of permanent magnet openings defined therein, forexample, permanent magnet openings 140, 142, and 144. A permanent magnetis positioned within each of permanent magnet openings 140, 142, and144, for example, a permanent magnet 150 is positioned within opening142. Opening 142 is sized so that each lamination used to form rotorcore 36 is manufacturable and movement of permanent magnet 150 withinopening 142 is reduced. In other words, openings 140, 142, and 144 areconfigured to restrict movement of permanent magnets positioned therein.Because of variability in the magnet manufacturing process, actualmagnet width and thickness may vary. There are also manufacturingtolerances introduced to rotor core 36 by the manufacturing process ofpunching magnet openings, for example, magnet openings 140, 142, and144, and stacking of laminations that form rotor core 36. Therefore,sufficient clearance is provided between a maximum magnet size and aminimum magnet opening size.

A ratio of a thickness 160 of permanent magnet 150 to a thickness 162 ofa bridge portion 164 between an outer circumference of permanent magnet150 nearest to outer surface 90 of rotor core 36 is equal to or greaterthan two. In the exemplary embodiment, rotor core 36 includes aplurality of rotor webs, for example, rotor webs 170 and 172. Rotor webs170 and 172 are portions of rotor core 36 that separate openings 140 and142, and openings 142 and 144, respectively. In the exemplaryembodiment, rotor webs 170 and 172 are substantially trapezoidal inshape. Rotor webs 170 and 172 are configured such that the laminationsused to form rotor core 36 may be manufactured using a standard punchingdie. Rotor webs 170 and 172 are designed and sized in such a way that aleakage flux, through rotor webs 170 and 172, is reduced. Reducingleakage flux increases motor efficiency by reducing the flux wasted asleakage flux, therefore, increasing the flux used by motor 10 to rotaterotatable assembly 18.

Each permanent magnet, for example, permanent magnet 150, includes anorth pole surface 174 and a south pole surface 176. In the exemplaryembodiment, the polarity of the permanents magnets within rotor core 36alternates between north pole surface 174 positioned facing outersurface 90 and south pole surface 176 facing outer surface 90. In orderto minimize leakage flux between north pole surface 174 and south polesurface 176 of permanent magnet 150, rotor webs 170 and 172 are thinenough to facilitate magnetic saturation of rotor webs 170 and 172. Incertain embodiments, the thickness of rotor webs 170 and 172 isconstrained by manufacturing capabilities. For example, a standardlaminate punching die may be capable of punching two openings separatedby a remaining section that is one to one and a half times the thicknessof the lamination. Typically, this remaining section between openingswould be too thick to facilitate magnetic saturation and thereforeminimize leakage flux. If rotor webs 170 and 172 are trapezoidal, thelarger end provides manufacturing integrity, while the smaller endfacilitates magnetic saturation of rotor webs 170 and 172.

FIG. 6 is a flow chart 200 of an exemplary method 210 for reducingcogging torque in an electric motor, for example, electric motor 10(shown in FIG. 1). Method 210 includes configuring 212 a rotor, forexample, rotatable assembly 18 (shown in FIG. 1) to rotate relative to astator, for example, stationary assembly 16 (shown in FIG. 1). Method210 also includes determining 214 an angle of a first end of the statortooth tip, for example, first end 100 of stator tooth tip 80 (shown inFIG. 2), relative to a center section of the stator tooth tip, forexample, center section 104 of stator tooth tip 80 (shown in FIG. 2).The angled surface results in a graded air gap, for example, air gap 96(shown in FIG. 2) when stationary assembly 16 (shown in FIG. 2) isassembled with rotatable assembly 18 (shown in FIG. 2). The graded airgap 96 reduces cogging torque to reduce vibration in electric motor 10and/or noise in a work component coupled to motor 10.

Determining 214 the angle of first end 100 of stator tooth tip 80relative to center section 104 of stator tooth tip 80 includesdetermining a first angle θ that is an angle measured from a first lineto a second line, for example, from first line 126 (shown in FIG. 4) tothird line 132 (shown in FIG. 4). First line 126 extends between anaxial center of the stator, for example, center 128 (shown in FIG. 4)and an intersection between center section 104 and first end 100. Thirdline 132 extends between center 128 and a point equidistant from firstend 100 of stator tooth 60 and an adjacent stator tooth. In theexemplary embodiment, the angle of first end 100 of stator tooth tip 80relative to center section 104 of stator tooth tip 80 is at leastpartially based on the first angle θ.

In the exemplary embodiment, first angle θ is determined by:

${0 < \theta \leq \frac{360^{\circ}}{2n}},$

where n=a number of stator teeth in electric motor 10.

Determining 214 the angle of first end 100 of stator tooth tip 80relative to center section 104 of stator tooth tip 80 comprisesdetermining a radial distance, for example, radial distance 136 (shownin FIG. 4). Radial distance 136 is defined as a difference between amaximum air gap and a minimum air gap between stator tooth tip 80 andthe rotor. The angle of the first end 100 of the stator tooth tip 80relative to the center section 104 of the stator tooth tip 80 isdetermined at least partially based on radial distance 136. In theexemplary embodiment, determining radial distance 136 includesdetermining a thickness of the center section, for example, distance 138(shown in FIG. 4). To reduce cogging torque, radial distance 136 isdetermined such that radial distance 136 is less than the thickness 138of center section 104.

Method 210 may also include determining 216 a profile of rotor webs, forexample, rotor webs 170 and/or 172 (shown in FIG. 5), included withinrotatable assembly 18. Rotor webs 170 and/or 172 configured tofacilitate reducing leakage flux through rotor webs 170 and/or 172. Theprofile of rotor webs 170 and/or 172 is determined to facilitatemagnetic saturation of rotor webs 170 and/or 172 and is at leastpartially dependent on manufacturing capabilities.

The methods, systems, and apparatus described herein facilitate reducingnoise in an application driven by a motor while maintaining, orminimally reducing, an efficiency of the motor. Electromagnetic designof a stationary assembly and a rotatable assembly used in an electricmotor facilitate reducing noise and increasing manufacturability, whileremaining cost effective. The methods, systems, and apparatus describedherein include a stator tooth shaped such that ends of tooth tips are ata larger radius than a center of the tooth resulting in a graded air gapwhen assembled with a rotor that is concentric to the stator. The gradedair gap reduces the amplitude of cogging torque at various frequenciesto reduce noise in the application.

The methods, systems, and apparatus described herein also facilitateefficient and economical assembly of a motor that includes an interiorpermanent magnet rotor. Permanent magnet openings within the rotor areconfigured to facilitate reducing leakage flux by allowing magneticsaturation of rotor webs within the rotor core. Permanent magnetopenings are also configured to account for manufacturing capabilities.Exemplary embodiments of the methods, systems, and apparatus aredescribed and/or illustrated herein in detail. The methods, systems, andapparatus are not limited to the specific embodiments described herein,but rather, components of each apparatus, as well as steps of eachmethod, may be utilized independently and separately from othercomponents and steps described herein. Each component, and each methodstep, can also be used in combination with other components and/ormethod steps. Furthermore, although described herein with respect to anelectric motor, the methods, systems, and apparatus described herein areapplicable to all electric machines, including electric motors andelectric generators.

Although specific features of various embodiments of the invention maybe shown in some drawings and not in others, this is for convenienceonly. In accordance with the principles of the invention, any feature ofa drawing may be referenced and/or claimed in combination with anyfeature of any other drawing.

When introducing elements/components/etc. of the systems and apparatusdescribed and/or illustrated herein, the articles “a,” “an,” “the,” and“said” are intended to mean that there are one or more of theelement(s)/component(s)/etc. The terms “comprising,” “including,” and“having” are intended to be inclusive and mean that there may beadditional element(s)/component(s)/etc. other than the listedelement(s)/component(s)/etc.

This written description uses examples to disclose the invention,including the best mode, and also to enable any person skilled in theart to practice the invention, including making and using any devices orsystems and performing any incorporated methods. The patentable scope ofthe invention is defined by the claims, and may include other examplesthat occur to those skilled in the art. Such other examples are intendedto be within the scope of the claims if they have structural elementsthat do not differ from the literal language of the claims, or if theyinclude equivalent structural elements with insubstantial differencesfrom the literal language of the claims.

What is claimed is:
 1. An interior permanent magnet machine comprising:a rotor rotatable about a central machine axis and comprising aplurality of permanent magnet openings and a plurality of permanentmagnets disposed therein; and a stator disposed coaxially with saidrotor and separated from said rotor by a circumferential air gap, saidstator comprising a plurality of stator teeth that define a plurality ofstator slots therebetween, said stator teeth comprising a stator toothtip configured to facilitate reducing cogging torque.
 2. A machine inaccordance with claim 1, wherein said air gap separates said rotor and asurface of said stator tooth tip, said stator tooth tip comprising afirst end, a second end, and a center section therebetween, said firstend located a first distance in a first circumferential direction fromsaid center section of said stator tooth tip and said second end locateda second distance in a second circumferential direction from said centersection of said stator tooth tip, the first circumferential directionopposite to the second circumferential direction.
 3. A machine inaccordance with claim 2, wherein said air gap is larger between saidrotor and said first end of said stator tooth tip than said air gap isbetween said rotor and said center section of said stator tooth tip. 4.A machine in accordance with claim 2, wherein said air gap is largerbetween said rotor and said second end of said stator tooth tip thansaid air gap is between said rotor and said center section of saidstator tooth tip.
 5. A machine in accordance with claim 2, wherein thesurface of said first end is angled relative to the surface of saidcenter section such that the air gap between said rotor and said statortooth tip increases as a circumferential distance from said centersection increases.
 6. A machine in accordance with claim 5, wherein theangle of said first section relative to the surface of said centersection is determined based at least partially on a frequency content ofa cogging torque generated by the machine.
 7. A machine in accordancewith claim 5, wherein the angle of said first section relative to thesurface of said center section is determined to facilitate reducing anamplitude of the cogging torque at at least one of a native frequency, apole pass frequency, a slot order frequency, and a stator winding backEMF harmonic frequency.
 8. A machine in accordance with claim 5, whereinsaid stator tooth tip is configured to at least one of reduce andmaintain an amplitude of the cogging torque at multiple harmonicfrequencies.
 9. A machine in accordance with claim 1, wherein saidpermanent magnet openings are separated by rotor webs configured tofacilitate reducing leakage flux through said rotor webs.
 10. A machinein accordance with claim 9, wherein a profile of said rotor webs issubstantially trapezoidal.
 11. A machine in accordance with claim 1,wherein said plurality of permanent magnets are centered within saidplurality of permanent magnet openings, said plurality of permanentmagnet openings configured to restrict movement of said plurality ofpermanent magnets disposed therein.
 12. A stator for use in an electricmachine, said stator comprising: an outer ring concentric about an axialcenter line of the electric machine; and a plurality of stator teethextending radially inward from said outer ring toward the axial centerline, each of said plurality of stator teeth comprising a stator toothtip having a first surface, said stator tooth tip comprising a firstend, a second end, and a center section therebetween, said first end andsaid second end taper away from the axial center line such that adistance between said first surface and the axial center line is greaterat said first end and said second end than at said center section, saidfirst surface configured such that an amplitude of a native coggingtorque fundamental is reduced, a higher harmonic back EMF is reduced,and a fundamental cogging torque at a pole pass frequency and a slotorder frequency is not substantially increased.
 13. A stator inaccordance with claim 12, wherein said first surface of said first endis angled relative to said first surface of said center section suchthat the distance between said first surface and the axial centerlineincreases as a circumferential distance from said center sectionincreases.
 14. A stator in accordance with claim 13, wherein the angleof said first surface of said first section relative to said firstsurface of said center section is determined based at least partially ona frequency content of a cogging torque wave generated by the electricmachine.
 15. A stator in accordance with claim 13, wherein the angle ofsaid first surface of said first section relative to said first surfaceof said center section is determined to facilitate reducing theamplitude of the cogging torque at at least one of the native frequency,a pole pass frequency, and a slot order frequency such that vibration issuppressed.
 16. A method for reducing cogging torque in an electricmachine that includes a rotor and a stator, said method comprising:configuring the rotor to rotate relative to the stator, the statorincluding a plurality of stator teeth each including a stator tooth tip;and determining an angle of a first end of the stator tooth tip relativeto a center section of the stator tooth tip that results in a graded airgap when assembled with the rotor, the graded air gap facilitatesreducing an amplitude of the cogging torque at a fundamental frequencywhile maintaining or reducing the amplitude of the cogging torque at atleast one other frequency.
 17. A method in accordance with claim 16,wherein determining an angle of a first end of the stator tooth tiprelative to a center section of the stator tooth tip comprisesdetermining a first angle θ that is an angle measured from a first lineto a second line, the first line extends between an axial center of thestator and an intersection between the center section and the first endand the second line extends between the axial center of the stator and apoint equidistant from the first end of the stator tooth and an adjacentstator tooth, wherein the angle of the first end of the stator tooth tiprelative to the center section of the stator tooth tip is at leastpartially based on the first angle θ.
 18. A method in accordance withclaim 17, wherein determining the first angle θ comprises:${0 < \theta \leq \frac{360^{\circ}}{2n}},$ where n=a number of statorteeth in the electric machine.
 19. A method in accordance with claim 16,wherein determining an angle of a first end of the stator tooth tiprelative to a center section of the stator tooth tip comprisesdetermining a radial distance, wherein the radial distance is defined asa difference between a maximum air gap and a minimum air gap between thestator tooth tip and the rotor, the angle of the first end of the statortooth tip relative to the center section of the stator tooth tip isdetermined at least partially based on the radial distance.
 20. A methodin accordance with claim 19, wherein determining the radial distancecomprises determining a thickness of the center section, wherein toreduce cogging torque and a higher harmonic back EMF, the radialdistance is determined such that the radial distance is less than thethickness of the center section.
 21. A method in accordance with claim16 further comprising determining a profile of rotor webs within theinterior permanent magnet rotor, wherein the rotor includes a pluralityof permanent magnet openings and a plurality of permanent magnetsdisposed therein, the permanent magnet openings separated by rotor websconfigured to facilitate reducing leakage flux through the rotor webs.22. A method in accordance with claim 21, wherein determining theprofile of rotor webs comprises determining the profile to facilitatemagnetic saturation of the rotor webs, the profile at least partiallydependent on manufacturing capabilities.